Which two of the following expressions represent the distance between [tex]\frac{3}{4}[/tex] and 2?

A. [tex]2 - \frac{3}{4}[/tex]
B. [tex]\frac{3}{4} - 2[/tex]
C. [tex]\left|2 - \frac{3}{4}\right|[/tex]
D. [tex]\left|\frac{3}{4} - 2\right|[/tex]



Answer :

To determine the distance between the fraction [tex]\(\frac{3}{4}\)[/tex] and the number 2, we can follow these steps:

1. Identify the Values:
- The first value is the fraction [tex]\(\frac{3}{4}\)[/tex].
- The second value is the number 2.

2. Calculate the Distance:
- The distance between two numbers on a number line is given by the absolute difference between them. This is represented by the absolute value function [tex]\( |\cdot| \)[/tex], which converts any number to its non-negative form.

3. Set Up the Absolute Difference:
- Subtract [tex]\(\frac{3}{4}\)[/tex] from 2: [tex]\( 2 - \frac{3}{4} \)[/tex].
- Alternatively, subtract 2 from [tex]\(\frac{3}{4}\)[/tex]: [tex]\(\frac{3}{4} - 2\)[/tex].

4. Apply the Absolute Value:
- Take the absolute value of each expression to ensure the result is non-negative:
[tex]\[ | 2 - \frac{3}{4} | \][/tex]
[tex]\[ | \frac{3}{4} - 2 | \][/tex]

5. Evaluate the Expressions (the distance calculation itself):
- Calculate the operations inside the absolute values:
[tex]\[ 2 - \frac{3}{4} = 2 - 0.75 = 1.25 \][/tex]
[tex]\[ \frac{3}{4} - 2 = 0.75 - 2 = -1.25 \][/tex]
- Convert each result to its absolute value:
[tex]\[ | 1.25 | = 1.25 \][/tex]
[tex]\[ | -1.25 | = 1.25 \][/tex]

So, the two expressions that represent the distance between [tex]\(\frac{3}{4}\)[/tex] and 2 are:
[tex]\[ | 2 - \frac{3}{4} | \][/tex]
[tex]\[ | \frac{3}{4} - 2 | \][/tex]

Both expressions evaluate to [tex]\(1.25\)[/tex], which correctly represents the distance between [tex]\(\frac{3}{4}\)[/tex] and 2.